The modal logic of continuous functions on cantor space

Abstract

Let $$\mathcal{L}$$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality $$\square$$ and a temporal modality $$\bigcirc$$ , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language $$\mathcal{L}$$ by interpreting $$\mathcal{L}$$ in dynamic topological systems, i.e. ordered pairs $$\langle X, f\rangle$$ , where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result.